Math Problem Statement
solve integration question
Solution
To find the area between the curves and , we need to determine the points where the curves intersect, then compute the integral of the difference between the two functions over the interval between those points.
Steps:
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Find the points of intersection: Set to solve for .
Factor the equation:
So, the points of intersection are and .
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Set up the integral: The area between the curves is given by:
Here, is the upper curve and is the lower curve.
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Compute the integral:
Now, evaluate the integral from 0 to 2:
Substituting :
Simplify:
Thus, the area between the curves is square units.
Would you like any further details or explanations? Here are 5 related questions to deepen your understanding:
- How do you determine which function is the upper or lower curve?
- What happens if the curves cross multiple times?
- How do you compute the area if the functions are not simple polynomials?
- What does it mean geometrically if the area is negative?
- How would the process change if we were asked to compute the volume between two surfaces instead of area?
Tip: When solving for areas between curves, always check the intersection points carefully to ensure you're integrating over the correct interval.
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Math Problem Analysis
Mathematical Concepts
Integration
Areas between curves
Intersection points
Formulas
Area between curves: A = ∫ (upper function - lower function) dx
Intersection points: Solve x^2 = 2x
Definite integral: ∫ (2x - x^2) dx = x^2 - (x^3)/3
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12 or College Calculus